A156305 G.f.: A(x) = exp( Sum_{n>=1} sigma(n) * C(2*n-1,n) * x^n/n ), a power series in x with integer coefficients.

1, 1, 5, 18, 87, 290, 1553, 5015, 25436, 94500, 431464, 1519749, 8024004, 26746757, 125190249, 498138920, 2221127601, 8020960187, 38836436844, 138444409552, 655009491676, 2512996318026, 10775473291178, 40824090856703

Offset: 0

Author

Paul D. Hanna, Feb 08 2009

Keywords

• nonn

Comments

Compare to g.f. of partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ),

and to the g.f. of Catalan numbers: exp( Sum_{n>=1} C(2*n-1,n)*x^n/n ),

where sigma(n) = A000203(n) is the sum of the divisors of n.

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 18*x^3 + 87*x^4 + 290*x^5 + 1553*x^6 + 5015*x^7 + ...
log(A(x)) = x + 3*3*x^2/2 + 4*10*x^3/3 + 7*35*x^4/4 + 6*126*x^5/5 + 12*462*x^6/6 + ... + A000203(n)*A001700(n)*x^n/n + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000203 (sigma), A000041 (partitions), A001700 (C(2*n-1, n)), A000108 (Catalan).

Programs

Mathematica

a[n_] := If[n==0, 1, (1/n) * Sum[DivisorSigma[1, k] * Binomial[2k - 1, k] a[n - k], {k, n}] ]; Table[a[n], {n, 0, 23}] (* Indranil Ghosh, Mar 12 2017 *)

PARI

{a(n)=polcoeff(exp(sum(k=1,n,sigma(k)*binomial(2*k-1,k)*x^k/k)+x*O(x^n)),n)}

PARI

{a(n)=if(n==0,1,(1/n)*sum(k=1,n,sigma(k)*binomial(2*k-1,k)*a(n-k)))}

Formula

a(n) = (1/n)*Sum_{k=1..n} sigma(k)*C(2*k-1,k)*a(n-k) for n>0, with a(0) = 1.